Famous anecdotes of G.H. Hardy relay that his work habits consisted of working no more than four hours a day in the morning and then reserving the rest of the day for cricket and tennis. Apparently his best ideas came to him when he wasn't "doing work." Poincare also said that he solved problems after working on them intensely, getting stuck and then letting his subconscious digest the problem. This is communicated in another anecdote where right as he stepped on a bus he had a profound insight in hyperbolic geometry.

I am less interested in hearing more of these anecdotes, but rather I am interested in what people consider an appropriate amount of time to spend on doing mathematics in a given day if one has career ambitions of eventually being a tenured mathematician at a university.

I imagine everyone has different work habits, but I'd like to hear them and in particular I'd like to hear how the number of hours per day spent doing mathematics changes during different times in a person's career: undergrad, grad school, post doc and finally while climbing the faculty ladder. "Work" is meant to include working on problems, reading papers, math books, etcetera (I'll leave the question of whether or not answering questions on MO counts as work to you). Also, since teaching is considered an integral part of most mathematicians' careers, it might be good to track, but I am interested in primarily hours spent on learning the preliminaries for and directly doing research.

I ask this question in part because I have many colleagues and friends in computer science and physics, where pulling late nights or all-nighters is commonplace among grad students and even faculty. I wonder if the nature of mathematics is such that putting in such long hours is neither necessary nor sufficient for being "successful" or getting a post-doc/faculty job at a good university. In particular, does Malcom Gladwell's 10,000 hour rule apply to mathematicians?

I think the last question you ask is very different from the others. In most of your question you appear to be talking about work done in grad school, but the 10,000 hour rule (if it is what I'm thinking of) applies to a very general sort of "work" done over one's entire life: thinking about mathematics! So I'm not sure which question you're asking.
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Qiaochu YuanDec 26 '09 at 23:54

7 Answers
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I agree that hard work and stubbornness are very important (I think we should all take after Wiles and Perelman as much as we can). But it is also important how you spend the many hours you dedicate to mathematics. For instance, choice of problems is quite important: it is important to make sure that when you work on something, you spend your time usefully, i.e. you not only make progress on this particular problem, but also learn something new about mathematics in general. It is also important not to get hyperfocused on a fruitless attempt to solve a problem; after some time and effort spent on it, it becomes addictive. In such a situation, it is sometimes better to stop and ask for help/read something or switch to another problem for a while. Often, you'll wake up one morning a month or a year later and see that the insurmountable obstacle has magically disappeared! Or maybe this "Aha!" moment will come during a discussion with another mathematician, or while listening to a talk. For many people it is also helpful to have many simultaneous projects, so that when you get stuck on one, you can work on another. To summarize, I think that not only the number of hours matters, but also how efficiently you spend them, not only in terms of publishable results,
but also in terms of your personal growth as a mathematician.

Neglecting good sleep as a way of doing mathematics is not a good idea. To do mathematics, you need to get enough rest. As to the number of hours per day, it is impossible to count this. When you think about a mathematical problem, you think about it all the time, including when you are asleep. It is true that mathematicians who work more do tend to achieve more, and those with the very top achievements do tend to work really a lot. But no amount of hours spent per day sitting at the table guarantees anyone anything in mathematics. The key word is not "long hours", but "dedication".

Well, I can't consider my self a successful mathematician, but from observing a few ones I know, here is my 2 cents:

The 10,000 hour rule (this is of course pseudo-science) certainly apply almost by definition:
most mathematician start training in college if not before. If you count up to postdoc, which is very typical, that is about 12 years, and an average of 3 hours/day give you 10,000 already.

Certainly work habits vary, but it does seem that quite a few successful mathematicians I have met know how to enjoy life. Having said that, I think a blessing/curse of our profession is that the lab is in our mind. So it looks like we do not work that hard, compared to some other fields, since some of my friends in physics/biology have to stay at the lab at nights frequently because of experiments. On the other hand, math can follow you around even when you are playing tennis (I can confirm that from personal experience!).

And it can be dangerous too! As a grad student, I fell off my bike once because I was distracted by thinking about mathematics. I broke my helmet and had to get stitches.
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Chris Schommer-PriesDec 31 '09 at 12:45

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You should probably just buy a new helmet. (Seriously, sorry to hear it, but any accident that causes your helmet to break must make you feel glad you were wearing it.)
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Pete L. ClarkFeb 1 '10 at 22:03

Grothendieck was known to work 10-12 hours a day more or less every day for the (something like) 25 years of his mathematical carrier. I think it's written in some memoirs of Cartier that this was one of the underlying reasons he decided to quit mathematics - he simply was too tired.

Every excellent mathematician that I know works extremely hard and for very long hours; however there are many others who also work extremely hard who are just average or "journeyman" mathematicians.

In other words, hard work is necessary but not sufficient - the existence of prodigies such as the Terry Taos and Akshay Venkateshs (just to name two Aussies that I've at least met) seems to me to be sufficient evidence that some natural talent/creativity/imagination/genius is necessary beyond mere hard work.

The working habits of mathematicians can also be endlessly amusing - we currently have a visitor who feels that he is full of creative energy immediately on waking, and so rather than getting out of bed and wasting that energy showering and having breakfast etc., he puts in an hour or two of maths and only starts the daily routine when he reaches his first dip in energy.

I think that your visitor has the right idea. I tend to make allowances for my post-lunch dip by doing paperwork or replying to e-mails. Whether it's changes in the level of ones cortisol, or just digestion it makes sense to plan for the day's highs and lows.
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James GriffinSep 16 '10 at 13:04

In order for your question to make sense, you need to define the terms "successful mathematician" and "good university". An idealistic response might skirt the issue altogether and claim that it does not matter what other people think. On the other hand, you are constantly being evaluated throughout your schooling and well into your professional career. Maybe you can focus on what it means to do good mathematics.

In his essay "What is Good Mathematics" Terrence Tao explains why he thinks Szemeredi's Theorem is good math. The theorem states that any subset of natural numbers with "positive upper density" contains arithmetic sequences of arbitrary length. Although this was proven by Szemeredi in the 1970's, it was proven by many other people using tools from all over mathematics: dynamical systems, Fourier analysis, hypergraph theory.

One of my favorite solutions is the proof of the Baik-Deift-Johansson conjecture on the longest increasing subsequence of a random permutation. Proofs of this theorem relate this statistic to eigenvalues of random Hermitean matrices and to the lengths of random Young tableaux under Plancherel measure. Again the techniques use here come from different branches of math: e.g. the Riemann-Hilbert correspondence, the representation theory of the symmetric group, orthogonal polynomials, random matrices and quantum gravity. See Longest Increasing Subsequences: From Patience Sorting to the Baik-Deift-Johansson Theorem

Good mathematics takes a certain mixed of creativity and technical know-how to pose and solve. By solving thousands of problems, you can develop your own mathematical taste. Then you can judge for yourself what good mathematics is and what it isn't.